"Image and Properties of Quadratic Functions" PPT Courseware 2

"Image and Properties of Quadratic Functions" PPT Courseware 2

Review goals:

1. Review and master the images and properties of quadratic functions.

2. Be proficient in finding the analytical expression of quadratic functions.

3. Master the relationship between quadratic functions, quadratic equations of one variable and quadratic inequalities of one variable.

Warm-up before class (students practice independently and make corrections in groups)

1. Three expression methods for the analytical expression of a quadratic function:

(1) General formula:___________

(2) Intersection formula:___________

(3) Vertex type:___________

2. Fill in the form:

3. Quadratic function y=ax²+bx+c, when a>0, on the right side of the symmetry axis, y increases with x as x increases, and on the left side of the symmetry axis, y increases as x increases with ____ _The graph has the most _____ point, and at this time the function has the most _____ value_____; when a<0, on the right side of the symmetry axis, y increases with x as x increases, and on the left side of the symmetry axis, As y increases with x, the _____ graph has a maximum _____ point, and the function has a maximum _____ value at this time.

Typical examples

Module 1: Opening direction, vertex coordinates, axis of symmetry, increase and decrease of parabola

1. The image of the function y=ax+1 and y=ax²+bx+1 (a≠0) may be ( )

2. It is known that the graph of the quadratic function y=ax²+bx+c is as shown in the figure. Try to judge the following symbols:

(1)abc__ 0 (2)b²-4ac__0

(3)2a+b__0 (4)a+b+c__0

This question mainly tests students' grasp of the graphics and properties of quadratic functions: the sign of b²-4ac depends on the intersection of the parabola and the x-axis; 2a+b looks on the position of the symmetry axis; and the sign of a+b+c should Look at the value of y when x=1.

Summary of rules

The sign of a——>Look at the opening of the parabola: the opening is upward, a>0; the opening is downward: a<0.

The sign of c——>Look at the intersection of the parabola and the Y-axis:

(1) The positive half-axis of the intersection Y-axis, c>0;

(2) The negative half-axis of the intersection Y-axis, c<0;

(3) Through the origin, c=0.

The sign of b——>Look at the axis of symmetry of the parabola: ______;

(Combined with the sign of a, you can determine the sign of b)

(1) If the axis of symmetry is on the right side of the y-axis, then ______ (different right);

(2) If the axis of symmetry is on the left side of the y-axis, then ______ (same as left);

(3) If the axis of symmetry is on the Y-axis, then ______.

The sign of b2-4ac——>Look at the intersection of the parabola and the x-axis:

1) If the parabola has two different intersection points with the x-axis: then b²-4ac>0;

2) If the parabola has only one intersection point with the x-axis: then b²-4ac=0;

3) If the parabola has no intersection with the x-axis: then b²-4ac<0;

The sign of a+b+c——>Look at the corresponding Y value on the image when x=1;

The symbol of a-b+c——>Look at the corresponding Y value on the image when x=-1;

Typical examples

Module 2 Translation of Quadratic Function

3. To get the image of the quadratic function y=-x²+2x-2, you need to change the image of y=-x² ( ).

A． Pan 2 units left and 2 units down

B． Pan right 2 units, then up 2 units

C. Move 1 unit left and 1 unit up

D. Pan right 1 unit, then down 1 unit

Module 3 Analytical expression of quadratic function

4. It is known that the graph of the quadratic function passes through the points (-2,0) (6,0), and the minimum value is -32. Find the analytical formula of the quadratic function.

y=2x²-8x-24

When it is known that the parabola passes through any three points, you can use the general formula, y=ax²+ bx+c (a≠0), and determine the values ​​of the coefficients a, b, and c.

When the vertex coordinates or axis of symmetry or maximum value of the quadratic function are known, the vertex formula y=a(x-h)²+k (a ≠ 0) can be used to determine the values ​​of a, h, and k.

When the intersection point of the parabola and the x-axis is known, or the length of the line segment intercepted on the x-axis, the intersection formula y=a (x-x1) (x-x2) (a ≠0) can be used to determine a, x1, value of x2.

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For more information about the "Graphics and Properties of Quadratic Functions" PPT courseware, please click the "Graphics and Properties of Quadratic Functions" ppt tag.

"Image and Properties of Quadratic Functions" PPT courseware 3:

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"Graphics and Properties of Quadratic Functions" PPT courseware:

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